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Pure Cubic Fields 1
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§ 4. Ambiguous principal ideals.
Since the cyclotomic unit group U_{k} is generated by < 1, ζ, >,
and since
Norm_{Nk}U_{N} ≥ Norm_{Nk}U_{k} = U_{k}^{3}
= < 1 >,
we obtain bounds for the unit norm index
(U_{k}:Norm_{Nk}U_{N}) ∈ {1,3},
according to whether ζ ∈ Norm_{Nk}U_{N} or not.
The somewhat abstract quotient E_{Nk}/U_{N}^{1  σ}
is isomorphic to the more ostensive quotient
P_{N}^{Gal(Nk)}/P_{k}
of the group of ambiguous principal ideals of Nk
modulo the subgroup of principal ideals of k,
according to Iwasawa (or also to Hilbert's Theorems 92 and 94),
and by the Takagi/Hasse Theorem on the Herbrand quotient, we have
(P_{N}^{Gal(Nk)}:P_{k}) =
(E_{Nk}:U_{N}^{1  σ}) =
3 (U_{k}:Norm_{Nk}U_{N}) ∈ {3,9}.
Even in the worst case (P_{N}^{Gal(Nk)}:P_{k}) = 3,
there are at least the radicals D^{1/3}, D^{2/3}, and the unit 1
which generate 3 distinct ambiguous principal ideals of LQ.
Examples 4.1.
We give the smallest radicands D of pure cubic fields L = Q(D^{1/3})
where the two values of #P_{r} = (P_{N}^{Gal(Nk)}:P_{k}) actually occur:
#P_{r} = 3 for D = 3 of type γ,
#P_{r} = 9 for D = 2 of type β.
§ 5. Differential principal factorizations (DPF).
As opposed to Hilbert's Theorem 94, the subgroup
P_{N} ∩ I_{k}/P_{k} ≤ P_{N}^{Gal(Nk)}/P_{k},
the socalled capitulation kernel of Nk is trivial, because h(k) = 1.
However, the elementary abelian 3group P_{r} = P_{N}^{Gal(Nk)}/P_{k},
whose generators are principal ideals dividing the different of Nk
(socalled differential principal factors),
consists of two nested subgroups, P_{r} ≥ P_{a}
(similar to but not identical with our definitions in [Ma2]),

absolute DPF of LQ, P_{a}, always containing the above mentioned radicals,

relative DPF of Nk, P_{r} \ P_{a},
such that the proper cosets of P_{a} in P_{r}
do not project down to LQ by taking the norm.
The existence, resp. the lack, of certain prime divisors of the conductor f
permits some criteria for the classification of pure cubic fields.
Theorem 5.1.

(U_{k}:Norm_{Nk}U_{N}) = 1
can occur only when f is divisible by no other primes than
3 and / or primes q_{j} ≡ ±1 (mod 9).

Relative DPF of Nk can occur only if
some prime q_{j} ≡ +1 (mod 3) divides f.

Absolute DPF of LQ, distinct from radicals, must exist when
no prime ≡ +1 (mod 3) divides f
and f has at least one prime factor distinct from
3 and from primes ≡ 1 (mod 9).
Remarks 5.1.
1. The condition for ζ ∈ Norm_{Nk}U_{N} is due to
the properties of the cubic Hilbert symbol (third power norm residue symbol) over k.
2. The condition for relative DPF of Nk is a consequence of the decomposition law
(e,f,g) = (1,1,2) for primes ≡ +1 (mod 3) in k.
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